Question

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Answer 1

$\int \frac{1}{3x^2+13x-10}dx$

$\mathrm{Take\:the\:partial\:fraction\:of}\:\frac{1}{3x^2+13x-10}:\quad \frac{3}{17\left(3x-2\right)}-\frac{1}{17\left(x+5\right)}$

$=\int \frac{3}{17\left(3x-2\right)}-\frac{1}{17\left(x+5\right)}dx$

$\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx$

$=\int \frac{3}{17\left(3x-2\right)}dx-\int \frac{1}{17\left(x+5\right)}dx$

$\int \frac{3}{17\left(3x-2\right)}dx=\frac{1}{17}\ln \left|3x-2\right|$

$\int \frac{1}{17\left(x+5\right)}dx=\frac{1}{17}\ln \left|x+5\right|$

$=\frac{1}{17}\ln \left|3x-2\right|-\frac{1}{17}\ln \left|x+5\right|$

$Add\:a\:constant\:to\:the\:solution$

$=\frac{1}{17}\ln \left|3x-2\right|-\frac{1}{17}\ln \left|x+5\right|+C$

Answer 2

kindly check the attachment for the stepwise solution and answer.

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